Abstract : The authors study the computational complexity of the discrete versions of some simple but basic decentralized decision problems. These problems are variations of the classical 'team decision problem' and include the problem of decentralized detection, whereby a central processor is to select one of two hypotheses, based on 1-bit messages from two non-communicating sensors. Results point to the inherent difficulty of decentralized decision making and suggest that optimality may be an elusive goal. (Author)